Skip to main content
SpringerLink
Log in

New Quantum Error-Correcting Codes from Hermitian Self-Orthogonal Codes over GF(4)

  • Conference paper
Finite Fields with Applications to Coding Theory, Cryptography and Related Areas

Abstract

In order to construct good quantum-error-correcting codes, we construct good Hermitian self-orthogonal linear codes over GF(4). In this paper we construct record-breaking pure quantum-error-correcting codes of length 24 with 2 encoded qubits and minimum weight 7 from Hermitian self-orthogonal codes of length 24 with dimension 11 over GF(4). This shows that length n = 24 is the smallest length for any known [[n, k, d]] quantum-error-correcting code with k ≥ 2 and d = 7. We also give a construction method to produce Hermitian self-orthogonal linear codes GF(4) from a shorter length such code.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Calderbank, A. R., Rains, E. M., Shor, P. W., Sloane, N. J. A. (1998) Quantum error correction via codes over GF(4). IEEE Trans. Inform. Theory. 44, 1369–1387

    Article  MathSciNet  MATH  Google Scholar 

  2. Gaborit, P., Huffman, W. C., Kim, J.-L., and Pless, V. (2001) On additive GF(4) codes. DIMACS Workshop on Codes and Association Schemes DIMACS Series in Discrete Math. and Theoret. Computer Science, American Mathematical Society, 56, 135–149

    MathSciNet  Google Scholar 

  3. Kim, J.-L. (2001) New self-dual codes over GF(4) with the highest known minimum weights. IEEE Trans. Inform. Theory. 47, 1575–1580

    Article  MathSciNet  MATH  Google Scholar 

  4. Shor, P. W. (1995) Scheme for reducing decoherence in quantum memory. Phys. Rev. A. 52, 2493

    Article  Google Scholar 

  5. Thangaraj, A., McLaughlin. S. W. (2001) Quantum codes from cyclic codes over GF(4m). IEEE Trans. Inform. Theory. 47. 1176–1178

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, 322 SEO(M/C 249), University of Illinois-Chicago, 851 S. Morgan, Chicago, IL, 60607-7045, USA

    Jon-Lark Kim

Authors
  1. Jon-Lark Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, 16802, University Park, PA, USA

    Gary L. Mullen

  2. FB6 Mathematik und Informatik, Universität Essen, 45117, Essen, Deutschland

    Henning Stichtenoth

  3. Departamento de Matemáticas, Universidad Autónoma Metropolitana, México, Izatapalapa 09340 Av. San Rafael Atlixco 186, 09340, México, D.F., México

    Horacio Tapia-Recillas

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kim, JL. (2002). New Quantum Error-Correcting Codes from Hermitian Self-Orthogonal Codes over GF(4). In: Mullen, G.L., Stichtenoth, H., Tapia-Recillas, H. (eds) Finite Fields with Applications to Coding Theory, Cryptography and Related Areas. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59435-9_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-59435-9_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63976-0

  • Online ISBN: 978-3-642-59435-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation